Monday, October 10, 2022

Liberals

 

You Don’t Own Me

You don't own me
Don't try to change me in any way
You don't own me
Don't tie me down 'cause I'd never stay

It is NOT being conservative to own the liberals.

My son’s POSSLQ, Person of the Opposite Sex Sharing Living Quarters, says that I am the biggest liberal she has ever met.  I found that shocking because I always thought that I was a conservative.  I believe in government but I think that it should be limited, not because it is evil, but because it is by necessity consists of people and according to Lord Acton’s dictum " power corrupts and absolute power corrupts absolutely".  I also believe that the government has a responsibility to act on behalf of the group, society, but it should carefully consider before undertaking any actions because of the Law of Unintended Consequences.  I also do not understand any exclusions from the group. My math background tells me that any growing society has to increase its variance, whose square root is standard deviation, not decrease deviation.  It also tells me that the future has to be worth less than the past in order for growth and not decay.

That makes me a conservative in the vein of Mitt Romney, not Matt Gaetz.  Like the voters in my home state of Massachusetts, I am most comfortable with a Democratic legislature and a Republican executive.  ( that is real Republicans who favor a republican form of government and not ”republicans  In Name Only” who are really authoritarians.)  If someone knows how to Razzle Dazzle, as much as I enjoy being Razzle Dazzled, I assume that they are using that Razzle Dazzle to distract me, not convince me.  Liberal and conservative alone is not a useful characterization.  It is much more nuanced than that.

Normal

A Wonderful Guy

I'm as corny as Kansas in August,
I'm as normal as blueberry pie.
No more a smart little girl with no heart,
I have found me a wonderful guy!

What does it mean to be normal?

The Cumulative Distribution Function, CDF, of an exponential distribution is an ideal function which has a value of zero at 0 and a value of 1 at infinity. However it is NOT a normal distribution, in that it has a skew of 2, as opposed to a normal distribution which has a skew of 0. An Exponentially Modified Gaussian Distribution has been proposed which combines the exponential and Gaussian (normal) distributions, but which results in a CDF with three parameters: µ, the mean; σ, the standard deviation; and λ, the rate parameter of the exponential distribution. A Gaussian distribution is NOT the only normal distribution, with a skew of 0 and where the mean, median and the mode are equal. Another normal distribution is the Logistics distribution which has two parameters: µ, the mean; and s, the scale parameter. However since s is a constant factor of of σ, s=√3/π * σ, it effectively has the same parameters as the Gaussian distribution. Reyes et al (Reyes, Venegas, & Gómez, 2018) proposed an Exponentially Modified Logistic distribution which has only two parameters : µ, the mean; and s, the scale parameter where the rate parameter of the exponential distribution is also equal to s.  However it is a solution where s, which is a constant factor of σ, is equal to λ, the rate parameter of an exponential distribution. It could have just as easily solved by setting µ equal to λ.

Figure 1 shows the Cumulative Distribution function, CDF, when σ, µ, and λ are all equal, in this case to 1. The CDF of the exponential function has the desired properties of being 0 when x is 0 and being 1 when x is infinity. Reyes’ Exponentially Modified Logistic distribution is closer to  normal distributions, whose CDFs are not zero when x is 0, but its CDF is closer to zero than that the CDF of a normal distribution such as a Gaussian or Logistic Distribution.

Figure 1 Cumulative Distribution functions, CDFs where µ=σ=λ


Figure 2 shows the CDFs when µ and λ are equal at 0.5, but the standard deviation σ, is twice that amount at 1.0. Also shown is the Exponentially Modified Gaussian CDF if the mean of an exponential distribution, 1/λ, is the equal to the mean of the Gaussian distribution, µ, as well as the Exponentially Modified Gaussian distribution if the standard deviation of an exponential distribution, 1/λ, is equal to the standard deviation of a Gaussian distribution, σ.


Figure 2 Cumulative Distribution functions, CDFs where µ=λ and σ=2*µ


Figure 3 shows the CDFs when σ and λ are equal at 0.5, but the mean µ is twice that amount at 1.0. Also shown is the Exponentially Modified Gaussian distribution CDF if the mean of an exponential distribution, 1/λ, is the equal to the mean of the Gaussian distribution, and the Exponentially Modified Gaussian distribution CDF if the standard deviation of an exponential distribution, 1/λ, is the equal to the standard deviation of the Gaussian distribution, σ.

Figure 3 Cumulative Distribution functions, CDFs where µ=λ and σ=2*µ



In all three figures, Reyes’ Exponentially Modified Logistics Distribution is closer to the shape of the CDF of normal distributions than the Exponentially Modified Gaussian distribution. When the parameters of the distributions are not equal, then the Exponentially Modified Gaussian Distribution does has a lower CDF when x is 0, but its does so by being less “normal” and closer to the shape of a skewed exponential distribution.

If a distribution is expected to be normal, then for its CDF to be close to the ideal function, it is suggested that Reyes’ Exponentially Modified Logistics Distribution be used. For this function to be normal, it appears to be more important that the standard deviation, σ, be equal to the inverse of the rate parameter of an exponential distribution, λ, than for the mean, µ, to be equal to the inverse of the rate parameter of an exponential distribution, λ.

 Reyes, J., Venegas, O., & Gómez, H. W. (2018). Exponentially-modified logistic distribution with application to mining and nutrition data. Appl. Math 12.6, 1109-1116.

  






Saturday, October 8, 2022

Functions

 

Walkin' on the Sun

And it ain't no joke when our mama's handkerchief is soaked With her tears because her baby's life has been revoked The bond is broke up, so choke up and focus on the close up Mr. Wizard can't perform, no God-like hocus-pocus

Where is Mr. Wizard when you need him? Perhaps I can invoke his memory in this blog post.

A function which has the value of zero at zero, and a value of 1 at infinity, describes a whole family of functions.  But if the inverse of that function should also have the value of 1 at zero and a value of zero at infinity, then that narrows it down considerably. And by that, it is meant an actual value of zero, not a discontinuity that is merely being defined as zero.  For example, the limit of f(x)=x as x approaches infinity is infinity, which means that the limit of 1/f(x) as x approaches infinity is zero, but this is a limit, NOT the actual value.  Thus, at x=0, there is a discontinuity for 1/f(x), i.e. 1/0, that is often defined as zero.  The function where there is no discontinuity will be a form of an exponential function.  The integral, Cumulative Distribution Function, of an exponential distribution,  f(x)=λ*exp(-λ*x), has a value of zero at zero and a value of 1 at infinity.  The integral of that distribution is an exponential association, 1‑exp(‑λ*x)

The exponential distribution does not allow values of x less than zero (and by that it is meant ABSOLUTE ZERO, not a relative zero which is merely a translation from absolute zero. E.g.  Centigrade/Celsius is a relative temperature scale where 0 degrees C is 273.15 degrees Kelvin, an absolute scale.).  An exponential distribution is NOT normal.  It has skew of 2 compared to a normal distribution with a skew of 0.  One combination of an exponential and Gaussian normal distribution which approaches normal is an exponentially modified Gaussian distribution.  It is defined by the exponential rate parameter, λ, AND the normal distribution parameters, σ, the standard deviation where the variance is the square of σ, and the mean, µ. 

Another normal distribution is the Logistic distribution.  That distribution is based on the probability of an individual member of a group, a distribution, selecting an alternative.  That probability itself is based on the mean, µ, of all probabilities, as well as the scale, s, range, over which that probability changes.  The scale, s, over which the probability changes is directly proportional to the standard deviation, σ.  Like other normal distributions, the mean of a logistic distribution is equal to its median. If the exponential distribution parameter, λ, is chosen to be equal to the scale parameter, s, and the exponential distribution is combined with the Logistic distribution then the result is as proposed by Reyes.  (Reyes, Venegas, & Gómez, 2018)  The exponentially modified Gaussian distribution and the exponentially modified Logistic distribution both also do not have the desired feature of zero at zero and infinity at infinity, but they are a step in that direction.

Figure 1 Cumulative Distribution Functions. 


The optimal desired function would also be one that is normal, e.g., has a skew close to zero as in the Gaussian or Logistics distribution, but allows for any value of x including negative values of x, unlike the exponential distribution, which appears to be a continuation of the exponential function on the y-axis with a limitation of x>0.  The desirable function is thus most probably a combination of a normal distribution and the exponential distribution.  While a feature of a normal distribution is that the mean is equal to the median, another feature of a normal distribution is that it follows the 68/95/99.7 rule; in other words,

68% of the values fall between the mean plus or minus 1 standard deviation, +/-σ;

95% of the values fall between the mean plus or minus 2 standard deviations, +/-; and

99.7% of the values fall between the mean plus or minus 3 standard deviations,+/- . 

Thus, if the skew is zero, 49.5% of the values, which is close to the median, 50% of the values, must fall between 0 and 3 standard deviations,- 3σ, from the mean, μ.  If the standard deviation, σ, is the square root of the variance and the skew is zero then, according to Pearson’s Second Coefficient of Skewness, the mean, μ, divided by 3 must be σ.  Thus, to be normal, have a skew of zero, the mean, μ, can not be more than 1.5 times the median. The exponential distribution, even though it has a skew of 2 compared to a normal distribution’s skew of zero, has a ratio of its mean to median of 1.44.  If the ratio of the mean to the median exceeds 1.5, the distribution is not only not normal, its observations can not consist of a single distribution.  Also, as the mean, μ, increases the variance, σ2, must also increase in order to be a normal distribution.  An exponentially modified Gaussian distribution has a range of the variance depending on the values, of μ, σ, and λ.  Within the valid ranges of the variance, the ratio of the mean to the median can not exceed 1.15.  An exponentially modified Logistics distribution has a range depending on the values, of μ, and s.  Within the valid ranges, the ratio of the mean to the median can not exceed 1.5.

If the observations are expected to represent a distribution of a phase change, i.e.  going from a cumulative probability of zero to a cumulative probability of 1, and the resulting equation is NOT required to be normal, then the regression equation should be expected to be the Cumulative Distribution Function of an exponential distribution, in other words, an exponential association. (i.e., the black line in Figure 1).  This requires only regressing, solving, for one parameter. That is, solving for the parameter λ by regressing the data to fit

y=1-e-λx

If the ideal equation is also expected to also be normally distributed, i.e., the equation should have a skew as close to zero as possible, then the cumulative Exponentially Modified Logistic Distribution should be used as the basis of the regression.  (i.e., the red line in Figure 1). This will require solving for two parameters: solving for µ and s, where s = √3/ π *σ, by regressing the data to fit

y=(1−e(μ-x) /s)*ln(e(x−μ) /s+1).

The ideal equation will be somewhere between the black line and the red line. 

It is noted that both regressions are of non-linear equations.  Linear regression is a more common technique.  It is possible through logarithmic transformation of the dependent, y, or independent, x, data to create some non-linear functions.

·        Linear-linear, where neither the y nor x data are transformed, under linear regression solves for

y = m*x +b; Linear

·        Log- linear, where the independent data, x, is transformed to be its natural logarithm and the dependent data, y, is not transformed, under linear regression solves for

y=b+m*ln(x); Logarithmic

·        Linear -Log where the independent data, x, is not transformed and the dependent data, y, is transformed to be its natural logarithm, under linear regression solves for

y= eb+m*x; Exponential

·        Log-Log, where both the y and x data are transformed to be their natural logarithm, under linear regression solves for

y=eb * xm; Power

While all but the first bullet are also non-linear equations, none of these bulleted equations are the non-linear equations shown in Figure 1.  This does not mean that if nonlinear regression software is not available, that linear regression can not be used.  It is only necessary to find the x observation at which the y data is 99.7% of the maximum value, and separate the data into buckets of equal intervals of the independent variable, x between 0 and that point.  In electrical engineering, the voltage of a resistor‑capacitor, RC, circuit over time follows an exponential association.   It has also been shown that segregating the data into six buckets and doing a linear regression on each bucket, can yield results which are linear regressions that approximate the nonlinear curve.  The same segregation and use of linear regression can be used in any application of exponential associations.  And as noted above, most group, distribution, behavior should be expected to be similar to that of an exponential association.  If that behavior is also expected to follow a normal distribution, it is a little more work to identify the parameters, but not insurmountable.

References

Reyes, J., Venegas, O., & Gómez, H. W. (2018). Exponentially-modified logistic distribution with application to mining and nutrition data. Appl. Math 12.6, 1109-1116.

 

 

 

 

 



Abortion II

 

Gimme Shelter

Rape, murder, it's just a shot away
It's just a shot away

Mmm, a flood is threatening
My very life today
Gimme, gimme shelter
Or I'm gonna fade away

Is abortion murder?

Abortion is the death of a life, that of the fetus.  But the fact that it is death does not mean that it is murder.  There are various causes of death, including murder.  When death is due to accident, disease or natural causes, we do not call it murder.  When a prisoner is executed by the state,  we call it Capital Punishment, not murder.  Even if the death is caused by another it may be ruled accidental, involuntary manslaughter, etc.  To be murder there has to be the intent to cause the death of another by an illegal means.

Being anti-abortion does not mean that one is pro-life.  If one is pro-life then any ending of life should be opposed, whether that death is from famine, war, pestilence or any of the other ills of the Four Horsemen, whose leader is Death and who fights on the side of evil in the Apocalypse.  Abortion may be death, but that does not make it Murder, any more than every copulation is Rape.  Gimme shelter.

Thursday, October 6, 2022

Love is Life

 

My Blue Heaven

There's a smilin' face, a fireplace, a cozy room
And a little nest that's nestled where the roses bloom;
Just Molly and me, and baby makes three,
We're happy in my Blue Heaven

Is three then the minimum number for Heaven?

I have argued in early posts that three is a special number, https://dbeagan.blogspot.com/2022/09/triads.html and https://dbeagan.blogspot.com/2022/09/third-parties.html.  In the same blog posts I noted that in game theory there are separate strategies for playing a two-person game by assuming the worst of the other player, while in three or more person games there are rules for continuing the game. It is not just in the political arena that this lesson might need to be learned. In order to make a baby it takes two, or at least that is what I think both religion and science say.

However, that two can be in a two-player game, and the act of conception is a dominance game, e.g. rape, incest, etc.; or it can be an act of love, e.g. Stevie Wonder’s marvelous song about the birth of his daughter…  isn't she lovely, made from love. Stevie Wonder in that same song goes on to say Life and love are the same. If you agree, and I hope that you do, this was a three-person game: Man, Woman and Life. If it is instead a two-person, dominance, game without love, then isn’t it death. So those approving of conception based not on love must be favoring death and forsaking Heaven, of any shade including Blue.

Monday, October 3, 2022

400 Posts

 

400 Years

Look how long four hundred years (four hundred years, four hundred years)
Way too long! (wo-o-o-o)
That's the reason my people (wo-o-o-o) my people can't see
Said, it's four hundred long years (four hundred years, four hundred years. wo-o-o-o)
Give me patience (wo-o-o-o) same philosophy

It hasn’t been 400 years, but it has been 400 posts

As I prepare for my 50th College anniversary,  and also looking over my blog posts, I am very grateful that 50 years ago I pursued a major which required 24 of the 28 courses needed for graduation. Given my blog posts, I might have  instead taken courses in:

In reality, I probably would have been like my senior year college roommate who had to, at the last minute, come up with a major that matched the courses that he had already taken. Too much variety is not always a good thing.

 

 

Student Loan Forgiveness

 

Busted.

I went to my brother to ask for a loan I was busted
I hate to beg like a dog for a bone but I'm busted
My brother said there ain't a thing I can do
My wife and my kids are all down with the flu
And I was just thinking of calling on you I'm busted!

Are you too busted to make or forgive a loan?

I received a Pell Grant to attend college and my household income is below the White House's threshold so I should be eligible for student debt relief.  But I have no dog in this hunt. My own student loans have long ago been repaid, which is hardly surprising because back in the 1970s, tuition at the Ivy League college I attended was only $1600 per semester, not the current over $25,000 per semester. However my children’s student loans have also been repaid.

An educated work force is an unpriced raw material that is required by most producers in a society. Public education has long been considered a public, government, function. Exceptions have been when religious or racial discrimination made public education not available to all members of society. This is why there are the parochial schools I attended before college or Historically Black Colleges and Universities. But, in general, public education has long been considered a proper government function. The fact that any of this education occurred after high school, does not make it any less a public unpriced  good. If it is an unpriced public good, there is no reason why there should have ever been any debt incurred by individuals to acquire this public good.

To those, including Evangelicals, who are disputing the wisdom of forgiving a mere portion of this debt, remember Jesus’s teachings

For the kingdom of heaven is like a landowner who went out early in the morning to hire laborers for his vineyard. After agreeing with the laborers for a denarius for the day, he sent them into his vineyard. When he went out about nine o’clock, he saw others standing idle in the marketplace, and he said to them, ‘You also go into the vineyard, and I will pay you whatever is right.’ So they went. When he went out again about noon and about three o’clock, he did the same. And about five o’clock he went out and found others standing around, and he said to them, ‘Why are you standing here idle all day?’ They said to him, ‘Because no one has hired us.’ He said to them, ‘You also go into the vineyard.’ When evening came, the owner of the vineyard said to his manager, ‘Call the laborers and give them their pay, beginning with the last and then going to the first.’ When those hired about five o’clock came, each of them received a denarius. Now when the first came, they thought they would receive more; but each of them also received a denarius. And when they received it, they grumbled against the landowner, saying, ‘These last worked only one hour, and you have made them equal to us who have borne the burden of the day and the scorching heat.’ But he replied to one of them, ‘Friend, I am doing you no wrong; did you not agree with me for a denarius? Take what belongs to you and go; I choose to give to this last the same as I give to you. Am I not allowed to do what I choose with what belongs to me? Or are you envious because I am generous?’ So the last will be first, and the first will be last.

I am planning on asking to be last myself. How about you?

 

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